Structured backward error for palindromic polynomial eigenvalue problems. II: Approximate eigentriplets
type: Item
instance of: scholarly article
title: Structured backward error for palindromic polynomial eigenvalue problems. II: Approximate eigentriplets
zbMATH Open document ID: 1447.65006
author: Changli Liu
author: Ren-Cang Li
published in: Frontiers of Mathematics in China
publication date: 2019-03-14T00:00:00Z
review text: Given an approximate eigentriplet for a palindromic polynomial matrix eigenvalue problem $P(\lambda)x = 0$, the paper tries to estimate the smallest perturbation of $P(\lambda)$ such that the given eigentriplet solves the perturbed problem exactly. It finds computable bounds for the structured backward errors where structured here means real symmetric or complex Hermitean matrices and similarly assumes only structured perturbations. This problem has applications in the vibration analysis of fast trains which invariably have both large and tiny magnitude eigenvalues. The authors' aim is to find the backward error of minimal norm that can be easily computed from the data. This optimization problem is first simplified via the QR decomposition of the given eigendata triple and then solved via detailed estimations of the thereby reduced number of variables. Numerical examples show the tightness of the found upper error bounds and two critical cases from earlier work on eigenpairs resurface here as well. The main complications for eigentriplets of palindromic matrix eigenproblems come from the doubling of coupled constraint equations in the optimization and bounding process when compared with the simpler and well understood eigenpair case. For Part I see [\textit{R.-C. Li} et al., Numer. Math. 116, No. 1, 95--122 (2010; Zbl 1213.65063)].
reviewed by: Frank Uhlig
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 65H17
Mathematics Subject Classification ID: 65G99
zbMATH DE Number: 7036356
zbMATH Keywords: eigentriplet
zbMATH Keywords: structured error analysis
zbMATH Keywords: matrix polynomial
zbMATH Keywords: polynomial eigenvalue problem
zbMATH Keywords: palindromic polynomial
zbMATH Keywords: error analysis
MaRDI profile type: Publication
full work available at URL: s11464-018-0738-4
OpenAlex ID: W2907914035
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Wikidata QID: Q128668444
DOI: 10.1007/S11464-018-0738-4
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description: scientific article; zbMATH DE number 7036356